is
|r1|2+. . .+|rn+1|2 = 1.
This is exactly the defining equation of then-dimensional sphereSnembedded inRn+1. We conclude that
O(n+ 1)/O(n)∼=Sn.
♣ Whenever we have a Klein geometry (G, H), we also have a pair of Lie algebras (g,h) where g is the Lie algebra of G and his the Lie algebra of H. Of course it follows from the definition of a Klein geometry that his a subalgebra ofg. This motivates the definition of a Klein pair.
Definition 2.2.3. A Klein pair (g,h) is a Lie algebra g with a Lie subalgebra h ⊂g. The kernelf of (g,h) is the largest ideal of gsuch thatf⊂h. Iff={0}we say that the Klein pair (g,h) is effective.
♠ While it is clear that any Klein geometry gives a Klein pair, it is not clear that there can be associated a Klein geometry to any Klein pair. In fact the later is not true, see [Sha00, Remark 3.8.10 and Definition 4.3.16]. It is however the case that effective Klein geometries provide effective Klein pairs. This follows from Lemma A.0.8in the appendix.
2.3 Principle Bundles and Principal Connections
Definition 2.3.1. Let ξ = (E, M, π, F) be a smooth fiber bundle. A group G makes ξ together with the right action E×G→E into a principal bundle if the right action is fiber preserving and acts simply transitively on each fiber, i.e. for each x, y ∈ F there exist a unique g∈Gsuch that xg=y.
♠ Notice that a simply transitive action is a transitive action with the additional property that the element g ∈ G such that xg =y is unique for any x, y ∈F. One could also make the equivalent demand that the action should be free and transitive, where free means that for any x∈F, ifxg=x theng=e, the identity element of G.
Lemma 2.3.2. A group action is simply transitive if and only if it is free and transitive.
Proof. If the action F ×G → F is simply transitive, then for any x ∈ F there is a unique g∈Gwithxg =x, butxe=x for all x∈F, henceg=eand the group action is free.
If the action is free and transitive, assume xg1 = xg2 for x ∈ F, g1, g2 ∈ G. Then xg1g−12 =x, and since the action is free this means g1g−12 = e, hence g1 = g2 which means the group action is simply transitive.
Example 2.3.3. Any Klein geometry (G, H) gives a natural principle bundle H → G → G/H. We already have a right action defined, that is the restricted group action ofH acting from the right onG. This action becomes transitive by construction. Also this action is free, since if we havegh=gfor some g∈Gand h∈H, theng−1gh=g−1g=e. In particular the cases from Example2.2.2are principle bundles:
(i) O(n)−→E(n)−→π Rn, (ii) O(n)→O(n+ 1)−→π Sn.
♣ Definition 2.3.4. LetP be a smooth manifold. LetHbe a Lie group and letµ:P×H→P be a smooth right action. Letµp :H →P be defined byµp(h) =µ(p, h), and then
µp∗:TeH−→TpP.
IfX∈h, the Lie algebra of H, then we define the vector fieldX]∈ T(P) onP by (X])p =µp∗(X)
♠ Notice that any principal bundle by definition has a smooth manifoldP with a Lie group H acting from the right, hence for any principal bundle we have a vector field X] for each X∈hwherehis the Lie algebra ofH. This is analogous to the left-invariant vector fields on a Lie group in some sense, as the example bellow will make clear.
Example 2.3.5. IfG is a Lie group andH ⊂Gis a subgroup with µ:G×H →Gdefined by µ(g, h) = gh, then we have a right action on a smooth manifold by a Lie group. In this case, notice that µg = Lg is just the left translation. Then, for any X ∈ h we have (X])g =µg∗(X) =Lg∗(X). In the special case wereH =Gwe see that this definition agrees with Definition 2.1.6. Thus we see that using the right action of a Lie groupG on itself we
can construct the left-invariant vector fields on G. ♣
Definition 2.3.6. Let P → M be a principal bundle with fiber G. A principal connection on P is a differential 1-form ω∈Ω1(P,g) with values in the Lie algebra g of Gsuch that
(i) Adg(R∗gω) =ω for all g∈G, where Rg is the right multiplication byg;
(ii) If X∈g and X] is the vector field onP associated to X, thenω(X]) =X.
♠ Example 2.3.7. In the trivial case whereM ={e}is a single point manifold, we get P =G a Lie group. Then the Maurer-Cartan form ωG on G would be a principal connection on G→ {e}. It is clear that the Maurer-Cartan form satisfy the second property, since in this case X] is exactly the left-invariant vector fields on G, see Example 2.3.5. That the first property is satisfied is a known result. To prove it we see that it is equivalent to
R∗gωG(V) =Adg−1ω(V).
for V ∈ T P. Look at Vp ∈ TpP and let X] be a left invariant vector field associated with X ∈g such thatXp] =Vp. On the left side we get
R∗gωG(Vp) =Rg∗ωG(X])
=ωG(Rg∗X])
= (Rg∗X])e
2.3 Principle Bundles and Principal Connections 13 and on the right side we get
Adg−1ωG(Vp) =Rg∗(Lg−1∗(ωG(X])))
=Rg∗(Xg]−1)
= (Rg∗X])e.
Since V was an arbitrary vector field on P and p an arbitrary point in P we can conclude that the Maurer-Cartan form is a principal connection onG→ {e}. ♣ Now, suppose (P, M, π, G) is a principal bundle over a smooth manifold M. Letp ∈ P, and consider
dπp:TpP −→Tπ(p)M.
Define Vp = ker(dπp)⊂TpP. The subspacesVp form a subbundleV ⊂T P called the vertical subbundle. Notice that Vp =Tp(π−1(m)), the tangent space to the fibre of π :P →M over m = π(p). But the fibers of the principal bundle is the Lie group G, hence Vp ∼=TgG ∼= g for some g ∈ G, where g is the Lie algebra of G. Notice that for any X ∈ g we have that X] ∈ V, hence any principal connection on P will look like the Maurer-Cartan form ωG on V in the sense of property (ii) of Definition 2.3.6 being fulfilled. This means that any principal connection ωp :TpP →gmust be a linear isomorphism when restricted to Vp. As a consequence, we could writeTpP =Vp⊕ker(ωp). Now we might think of defining a connection on a principal bundle by choosing a subbundle of the tangent bundle of the principal bundle H ⊂T P such thatT P =H ⊕ V. Some literature use this approach when defining a principal bundle. We give a definition similar to [Joy09, Def.2.1.6].
Definition 2.3.8. Let P → M be a principal bundle with fiber G. A prncipal Ehresmann connection onPis a vector subspaceH ⊂T P called the horizontal subbundle, that is invariant under theG-action onP, and which satisfies TpP =Vp⊕ Hp for each p∈P.
♠ The statement thatHis invariant under the action ofGonP means thatRg∗(Hp) =Hpg for allp∈P,g∈G. Notice that dπp is a linear map that maps TpP ontoTπ(p)M, and since Vp = ker(dπp) we have an isomorphism Tπ(p)M ∼= Hp. It is worth noting that a principal Ehresmann connection is a special case of the more general Ehresmann connection that could be defined on any fiber bundle. We shall see bellow that there is a one-to-one correspondence between principal connections and principal Ehresmann connections. This means that the two definitions are equivalent in some sense, and we might choose to work with one or the other.
Lemma 2.3.9. There is a one to one correspondence between principle connections and principle Ehresmann connections.
Proof. We start by showing that a principal Ehresmann connectionH ⊂T P induces a prin-cipal connection. Letφ:T P → V be a bundle morphism, i.e. a fiber preserving continuous map considering bothV and T P as fiber bundles overM, such that
• φ(φ(v)) =φ(v) for allv∈T P,
• φ|V =idV, the identity map on Vp.
Such a projection is uniquely determined by its kernel, hence any principal Ehresmann con-nectionH ⊂T P induces a projectionφ:T P → V by ker(φ) =H. Let ı :V −→g be a map such thatıp(Xp]) =X∈g. Define theg-valued 1-form
η =ı◦φ:T P −→g.
We need to show that this 1-form satisfies the two properties that makes it a principal connection:
(i) Notice that the adjoint mapAdg =d(Rg−1 ◦Lg), so for anyX ∈g we have Adg(X) =d(Rg−1 ◦Lg)(X)
=Rg−1∗(Lg∗(X))
=Rg−1∗(X)
since the Lie algebra can be represented as the left-invariant vector fields on G. It remains to be shown that Rg−1∗(η(Rg∗(v))) = η(v) for any v ∈ T P. We can write v=vH+vV wherev∈TpP,vH∈ HpandvV ∈ Vp. Thenη(v) =ıp(φ(vH+vV)) =ıp(vV).
What we end up with is
Rg−1∗(η(Rg∗(v))) =Rg−1∗(η(Rg∗(vH+vV)))
=Rg−1∗(η( ˜vH+Rg∗vV))
=Rg−1∗(ı(Rg∗(vV)))
=ı(vV)
=η(v)
(ii) Let the vector fieldsX]be constructed by using the right actionµ:P×G→P. Recall that this right action is fiber preserving. That means that, using the local trivialization of P, the map
µp∗ :g−→TpP ∼=TmM⊕TaG
can be evaluated as τ(µp∗(X)) = (OTmM, Ra∗X)∈TmM⊕TaG, where p= (m, a) with m = π(p) ∈ M and a ∈ G, and τ is the map of the local trivialization. Going back to TpP, recall that Vp ∼= g ∼=TaG and Hp ∼= TmM, hence Xp] =µp∗(X) ∈ Vp for each p∈P,X ∈g.
To show the other direction, letω be a principal connection onP, and define H ⊂T P as H= ker(ω).
We need to show that
(i) TpP =Hp⊕ Vp for all p∈P;
(ii) Rg∗(Hp) =Hpg for allp∈P and g∈G.
2.4 Cartan Geometry 15